Theoretical Mathematics & Applications, vol.3, no., 013, 99-115 ISSN: 179-9687 (print), 179-9709 (online) Scienpress Lt, 013 Lagrange s equations of motion for oscillating central-force fiel A.E. Eison 1, E.O. Agbalagba, Johnny A. Francis 3,* an Nelson Maxwell Abstract A boy unergoing a rotational motion uner the influence of an attractive force may equally oscillate vertically about its own axis of rotation. The up an own vertical oscillation will certainly cause the boy to possess another ifferent generalize coorinates in aition to the rotating coorinate. We have shown analytically an qualitatively in this wor, the effect of the vertical oscillating motion of a boy cause by the vibrational effect of the attractive central force. The total energy possess by the boy is now the sum of the raial energy an the oscillating energy. The results show that the total energy is negative an highly attractive. Keywors: Elliptical plane, vertical oscillation, critical velocity 1 Department of Physics, Feeral University of Petroleum Resources, Effurun, Nigeria. Department of Physics, Feeral University of Petroleum Resources, Effurun, Nigeria. 3 Bayelsa State College of Eucation Opoama, Brass Islan Bayelsa State. * Corresponing Author. Article Info: Receive : March, 013. Revise : May 6, 013 Publishe online : June 5, 013
100 Lagrange s equations of motion for oscillating central-force fiel 1 Introuction A central force is a conservative force [1]. It is a force irecte always towar or away from a fixe center O, an whose magnitue is a function only of the istance from O. In spherical coorinates, with O as origin, a central force is given by F f ( r) rˆ. Physically, such a force represents an attraction if ( f ( r) 0) an repulsion if ( ( r) 0) r 0. f, from a fixe point locate at the origin Examples of attractive central forces are the gravitational force acting on a planet ue to the sun. Nuclear forces bining electrons to an atom unoubtely have a central character. The force between a proton or an alpha particle an another nucleus is a repulsive central force. The relevance of the Central - force motion in the macroscopic an microscopic frames warrants a etaile stuy of the theoretical mechanics associate with it. So far, researchers have only consiere central - force motion, as motion only in the translational an rotational plane with coorinates ( r, θ ), for example, see Keplerian orbits [, 3]. However, the theoretical nowlege avance by these researchers in line with this type of motion is scientifically restricte as several possibilities are equally applicable. There exist four stanar formulations of classical mechanics: (i) Isaac Newton s formulation Newtonian mechanics (ii) Lagrange s formulation Lagrange s mechanics (iii) Hamilton s formulation Hamiltonian mechanics (iv) De Alambert s formulation De Alambertian mechanics. All these formulations are utilize in the theory of mechanics where applicable. Some of the conitions satisfie by a boy unergoing a Central - force motion is as follows: (i) the motion of the boy can be translational an rotational in the elliptical plane with polar coorinates ( r, θ ), (ii) the boy can be rotating an revolving about its own axis in the elliptic plane ( r, θ ), (iii) the boy can be translating an rotating in the elliptical plane ( r, θ ), at the same time, oscillating
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 101 up an own about its own axis (iv) the boy can be translating an rotating in the elliptic plane ( r, θ ), at the same time, oscillating up an own above the axis of rotation but not below the axis of rotation (v) the combination of any of these conitions form another class of a central - force motion. In orer to mae the mechanics of a Central - force motion sufficiently meaningful, we have in this wor extene the theory which has only been that of translational an rotational in the elliptical plane with polar coorinates ( r, θ ), by incluing spin oscillation. Uner this circumstance, we shall be contening with a total of 6 - generalize coorinates or egrees of freeom; from the translational an rotational motion in the elliptical plane ( r, θ ), from the orbital spin oscillations ( β, α) an from the tangential spin oscillations ( µ, φ). Consequently, these parameters form the basis of our classical theory of 6-imensional motion. The number of inepenent ways in which a mechanical system can move without violating any constraints which may be impose is calle the number of egrees of freeom of the system. The number of egrees of freeom is the number of quantities which must be specifie in orer to etermine the velocities of all particles in the system for any motion which oes not violate the constraints [4]. There is a single source proucing the force that epens only on istance in the theory of central-force motion an the force law is symmetric [5]. If this is the case, then, there can be no torques present in the system as there woul have to be a preferre axis about which the torques acts. In this wor, we are solving the problem of oscillating central force motion in a resistive non-symmetric system. That is, the upwar isplacement is not equal to the ownwar isplacement in the tangential spin oscillating phase. Consequently, the raii istances from the central point are not equal. This however, causes torques thereby maing the system uner stuy non-spherically symmetric.
10 Lagrange s equations of motion for oscillating central-force fiel Meanwhile, I hereby request the permission of the reaer to excuse the lac of intensive references to the current literature. I on t now of other current authors who have stuie these questions before now. I believe this is the first time this wor is uner investigation. This paper is outline as follows. Section 1, illustrates the basic concept of the wor uner stuy. The mathematical theory is presente in section. While in section 3, we present the analytical iscussion of the results obtaine. The conclusion of this wor is shown in section 4 an this is immeiately followe by appenix an list of references. Mathematical theory.1 Evaluation of the velocity an acceleration We have elaborately shown in (A. 6) in the appenix that the position vector r of a boy whose motion is translational an rotational in a plane polar orbit as well as oscillating about a given equilibrium position in a central-force motion is given by the equation r r rˆ r rˆ( θ, β, µ, α, φ) (.1) r v t r rˆ θ rˆ β rˆ µ rˆ α rˆ φ rˆ + r + + + + t θ t β t µ t α t φ t (.) v r rˆ + r θ ˆ θ + r β ˆ β + r µ ˆ µ + r α ˆ α + r φ ˆ φ (.3)
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 103 r v rˆ θ rˆ β rˆ µ rˆ α r ˆ ϕ a rrˆ + r + + + + t t θ t β t µ t α t ϕ t ˆ ˆ ˆ ˆ θ ˆ ˆ β + r θθ + r θθ+ r θ + r ββ+ r ββ+ r β + r µµ ˆ θ β ˆ µ ˆ α ˆ ˆ ˆ ϕ + r ϕϕˆ+ rϕϕˆ+ rϕ µ α ϕ + rµµ + rµ + rαα + rαα + rα ˆ ( θ ) ˆ ( θ θ) ˆ θ ( β β β tan β) ˆ β + ( r µ + r µ r µ tan µ r µ cot µ ) ˆ µ + ( r + r r tan ) ˆ + ( r + r r tan r cot ) a r r r+ r + r + r + r r α α α α α ϕ ϕ ϕ ϕ ϕ ϕ ˆ ϕ (.4) (.5) while the symbols appearing in (.1) - (.5) have been clearly efine in the appenix. However, β is the upper raial orbital oscillating angle an α is the lower raial orbital oscillating angle. Note that both of them are projections of the tangential oscillating plane onto the orbital elliptical plane. However, let us isengage the acceleration equation in (.5) with the view that the 5 th an the 8 th terms have the elements of angular momentum an the orbital oscillating phases. Thus ( ) ˆ ( ) ( ) ( tan ) + ( r µ + r µ r µ tan µ r µ cot µ ) ˆ µ + ( r α + r α) ˆ α ( r α tanα) ˆ α + ( r ϕ+ r ϕ r ϕ tanϕ r ϕ cotϕ) ˆ ϕ a r rθ r+ rθ + rθ ˆ θ + rβ + rβ ˆ β rβ β ˆ β (.6) Equation (.6) is now the new acceleration equation which governs the motion of a boy unergoing a central-force motion when the effect of vertical oscillation is ae.
104 Lagrange s equations of motion for oscillating central-force fiel. Evaluation of the central- force fiel In classical mechanics, a central force is a force whose magnitue only epens on the istance r, of the boy from the origin an is irecte along the line joining them [5]. Thus, from the analytical geometry of the central-force motion shown in Figure A. 1, in the appenix, permits us to write in terms of vector algebra that F ( r) f ( r )(ˆ; r ˆ, β ˆ) α f ( r ) rˆ + f ( r ) ˆ β + f ( r ) ˆ α m a (.7) where F is a vector value force function, f is a scalar value force function, r is the position vector, r is its length, an r ˆ r / r, is the corresponing unit vector. We can convert (.6) to force by simply multiplying it by the mass m of the boy an equate the resulting expression to (.7). Note that we are utilizing the orbital oscillating phase in (.6), which is acting raially in the irections of βˆ anαˆ in our calculation. Once this is one, we obtain the following sets of canonical equations of motion. {( r r θ ) ( r β tan β r α tanα )} ( r θ + r θ ) 0 f ( r) m + (.8) m (.9) ( r µ + r µ ) 0 m (.10) ( r β + r β ) 0 m (.11) ( r α + r α ) 0 m (.1) ( r φ + r φ) 0 m (.13) The sets of canonical equation (.9)-(.13) etermines the angular momentum which are constants of the motion acting in the irections of increasing coorinate, θ, β, µ,α an φ. Equation (.8) is the require new central-force fiel which we have evelope in this stuy. It governs the motion of a boy uner a central-force when the effect of vertical spin oscillation is ae.
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 105.3 Evaluation of the oscillating energy E osc in the tangential phase There is no force acting in the irection of the orbital angular acceleration, the lower an upper tangential oscillating phase. Since the force acting is a central force, it is always in the irection of the raial acceleration. The orbital angular acceleration, the lower an upper tangential oscillating phase are perpenicular to the line OP an it is in the increasing orer ofθ, µ anφ. As a result, by converting the tangential oscillating phase of the acceleration to force an equate the result to zero, we get ( r µ + r µ r µ tan µ r µ cot µ ) 0 m (.14) ( r φ + r φ r φ tanφ r φ cotφ) 0 m (.15) Because of the similarity in the two equations, we shall only solve (.14) an assume the same result for the other one. Hence from (.14) m r t t ( r µ + r r µ r µ tan µ r µ cot µ ) 0 ( r µ ) ( r µ tan µ + r µ cot µ ) 0 t {( r µ ) ( r µ tan µ + r µ cot µ ) } 0 ( r µ tan µ + r µ ) EI ; ( r 0) (.16) m (.17) (.18) r µ cot µ (.19) an with a similar equation for (.15) in frame II as r φ ( r φ tanφ + r φ cotφ) EII (.0) E E + E (.1) osc I ( µ + φ) r ( µ tan µ + φ tanφ) r ( µ cot µ + φ cot ) E osc r φ (.) The oscillating energy is a function of the raius vector an it increases negatively as the vertical oscillating angles are increase. Hence, the oscillating II
106 Lagrange s equations of motion for oscillating central-force fiel energy posses by the boy in terms of µ an φ in the oscillating phase is given by (.). This equation etermines how energy is conveye up an own in the vertical oscillating phase..4 Relationship between the raial velocity an the tangential oscillating angles To etermine the tangential oscillating angles we consier (.14) an assume possibly that for m 0 ( µ + r µ r µ tan µ r µ cot µ ) 0 r (.3) + 4r µ ( tan µ + cot µ ) ( tan µ + cot µ ) r ± 4r µ (.4) r r r ± r 1+ µ ( tan µ + cot µ ) r µ (.5) r ( tan µ + cot µ ) The iscriminate of (.5) is zero provie ( tan µ + µ ) ( tan µ + µ ) r r µ cot (.6) r r µ cot (.7) Similarly, by following the same algebraic proceure for (.15), we obtain an r r ± r 1+ φ ( tanφ + cotφ) r φ (.8) r ( tanφ + cotφ) ( tan φ + cotφ) r r φ (.9) Thus the raial velocity is irectly proportional to the raius vector an irectly proportional to the square root of the vertical oscillating angles. The raial velocity ecreases as the vertical spin oscillating angles are increase.
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 107.5 Evaluation of the Lagrange s equations of motion From equation (.3) we realize that the inetic energy T of the boy can be written as 1 1 ( r + r θ + r β + r µ + r α r φ ) T m v m + (.30) L T V (r) (.31) t L q L q 0 (.3) t q q ( T V ( r) ) ( T V ( r) ) 0 (.33) t T q V ( r) T q q V ( r) + q 0 (.34) q ( V ( r) ) 0 ; V ( r) V q q (.35) where counterparts, t T q T q V ( r) + 0 q q ( r, θ, β, µ, α, φ ) ; ( r, θ, β, µ, α, φ ) q are the generalize coorinates, (.36) q (.37) q are the associate velocity L / q generalize velocity, L / q generalize momentum. Remember that the only requirement for the generalize coorinates is that they span the space of the motion an be linearly inepenent. Since the force is raially symmetric, let us evaluate (.36) first with respect to the generalize coorinate q r. Then the Lagrange s equations of motion is t V ( r) (.38) r ( m r) m r ( θ + β + µ + α + φ ) + 0
108 Lagrange s equations of motion for oscillating central-force fiel t ( m r) m r ( θ + β + µ + α + φ ) f ( r) 0 (.39) Also from (.36), since L an V (r) are not functions of the generalize coorinates q, then we have T q an as a result (.36) becomes, V ( r) q 0 (.40) t T q 0 (.41) Hence, after some straightforwar algebra we obtain the following generalize momenta. t t ( m r ) θ 0 ( m r β ) 0 ; ( mr θ ) l ; ( m r ) l β l θ (.4) mr ; l β (.43) m r ; t t t ( m r µ ) 0 ( m r α ) 0 ( m r ) φ 0 ; ( m r ) l µ ; ( m r ) l α ; ( m r ) l φ l µ (.44) m r ; l α (.45) m r ; l φ (.46) m r ; The canonical set of equations given by (.4) (.46) are referre to as the Lagrange s equations of motion for the boy of mass m. Suppose we now replace (.4) (.46) into (.39) so that we realize t l l l l l 4 4 4 4 4 (.47) m r m r m r m r m r ( m r) m r + + + + f ( r) 0 t l (.48) mr ( m r) 5m r f ( r) 0
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 109 t t 5l (.49) 3 mr ( m r) f ( r) 0 5l (.50) 3 mr ( m r) r r f ( r) r 0 5l ( m r ) r f ( r) r t 0 (.51) 3 mr 1 5l r 0 t (.5) m r m r f ( r) r 3 1 5l m r + f r r E r m r ( ) (.53) Equation (.53) gives the raial energy posses by the boy as it oscillates tangentially an translates rotationally roun the central point. The equation provies the energy of the boy in terms of the translational raial velocity, the angular momentum an the raial force. The reaer is irecte to (A.) in the appenix, where the factor of half which appears in (.5) is iscusse. However, for a conservative fiel V ( r) f ( r) r (.54) from which 5l r E V ( r) (.55) m m r mr m l E V ( r) r (.56) 5 It is evient from (.55) that the translational raial velocity of the boy epens only upon the raius vector. This of course efines the thir property of central force motion. The translational raial velocity is etermine by the energy as a constant of the motion, the effective potential an the angular momentum. Therefore, the total energy E t possess by the boy is now the sum of the raial energy E an the oscillating energy E. r osc
110 Lagrange s equations of motion for oscillating central-force fiel 1 5l E t m r + f ( r) r + (.57) m r ( µ + φ) r ( µ tan µ + φ tanφ) r ( µ cot µ + φ cotφ) r (.58) 3 Discussion of results The oscillating energy E osc is mae up of three inepenent generalize coorinates an two major parts. The first part is the vertical spin oscillating velocities which is perpenicular to the irection of the raius vector. The secon part in (.) is the unboune oscillating phase. The unrestricte nature of the integrals of E osc, means that the oscillating phase has several possibilities of oscillation. However, the secon term of E osc increases as the vertical spin oscillating angles is increase. Whereas, the thir term ecreases as the vertical spin oscillating angles is increase. The total energy angular momentum part of E t comprises of the raial an the oscillating part. The E t has a higher appreciable value compare to the usual equation of central-force motion an a negative effective potential. The last integran of E t becomes negatively small an negligible as the vertical spin oscillating angles is increase. 4 Conclusion In general, we have in this stuy solve the problem of the motion of a boy in a plane polar coorinate system that is subject to a central attractive force which is nown an, in aition, a rag oscillating force which acts tangentially. The oscillating energy Eosc which etermines how energy is conveye up an own in
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 111 the oscillating phase is relatively etermine by the vertical spin oscillating angles. The new force law now comprises of the raial an the tangential oscillating parts which reuces the strength of the attractive central force fiel. The nowlege of this type of central force motion which we have investigate in this wor can be extene from plane polar coorinate system to that of spherical an cylinrical polar coorinate systems. Appenix Let us consier the rotational motion of a boy of mass m about a fixe origin say, O, in an elliptic polar coorinate ( r, θ ) system. Suppose the boy is also oscillating up an own about its equilibrium position as it translates rotationally roun the fixe origin. The boy thus possesses translational an rotational elliptical motion with polar coorinates ( r, θ ) an tangential spin oscillating motion escribe by the vertical isplacement C D C B C an repeately in the y -irection. The geometry of the analytical requirements is shown in Figure A. 1. The reaer shoul tae note that the oscillation of the boy is not entirely out of the elliptical orbit of rotation. Rather the isplacement D an B above an below C is very small. The oscillation is still within the limits of the axis of rotationc. We have only ecie to stretch D an B above an below C consierably enough in orer to reveal the geometrical concept require for the analytical calculation. There are six possible egrees of freeom or generalize coorinates exhibite by the motion boy uner this circumstance: (i) translational an rotational in the elliptical plane ( r, θ ), (ii) the plane of upwar oscillations ( β, µ ) an (iii) the plane of ownwar oscillations ( α, φ).
11 Lagrange s equations of motion for oscillating central-force fiel We shall compute separately the tangential spin oscillating motions in both oscillating frames an eventually combine the result with the orbital elliptical plane motion. In this stuy, we assume that the angular isplacements in the tangential spin oscillating frames are not equal an so the system uner stuy is not raially symmetric. Consequently, there is the existence of torque ue to the non uniformity of the raii istances. Accoringly, we can now evelop relationships between the various areas inicate on Figure A. 1, with the goal to fin the formula for the area swept out by the elliptical plane polar motion, an the result obtaine from this is then ae to the tangential oscillating triangle sections DO ˆ C an CO ˆ B respectively. From the figure, P an Q are very small upwar an ownwar isplacements from the equilibrium axis of rotationc, that is, regions in the upper an lower triangular swept segments of the upper an lower elliptical plane. Our first tas woul be to connect all these oscillating spin angular egrees of freeom into an expression in terms of P an Q. For clarity of purpose, let us efine the various symbols which we may encounter in our calculations : (i) the elliptical raius r (ii) the plane of upwar oscillations ( β, µ ), that is subtene from the upper elliptical plane (iii) the plane of ownwar oscillations ( α, φ), that is subtene from the bottom or the lower part of the elliptical plane (iv) the elliptical orbital angle θ (vi) the upper tangential oscillating spin angle µ (v) the lower tangential oscillating spin angle φ (vi) the upper an lower orbital spin oscillating angles β an α
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 113 Figure A.1: Represents the elliptical an oscillating motion of a boy in a centralforce fiel. The boy is oscillating up an own about the axis of rotationc. Where DO ˆ C (frame I) an COˆ B (frame II) are the upper an lower projections onto the plane of the ellipse, line DC ( P ) an CB (Q ) are very small isplacements from the axis C, we have only stretche them to mae the geometry of the figure clear enough for the calculation. However, β is the upper orbital oscillating angle an α is the lower orbital oscillating angle. Note that both of them are projections of the tangential oscillating plane onto the orbital plane of the ellipse. In frame I : we obtain from DO ˆ C r1 r cos ecµ ; P r sin β r sin β cos ecµ (A.1) In frame II : we obtain from CO ˆ B 1 r r cos ecφ ; Q r sinα r sinα cosecφ (A.) In the orbital plane of rotational an translational motion, the position vector r of the boy is given by r xi r cosθ i (A.3) However, the combination of the rotational an translational motion, with the vertical spin oscillating frames (acting in the y -irection), will yiel
114 Lagrange s equations of motion for oscillating central-force fiel ( cosθ sin β cos µ sinαcos ϕ ) r xi + y j xi + Pj + Q j r i + ec j + ec j (A.4) r rˆ r ( cos θ i + sin β cosecµ j ) + sinα cosecφ j (A.5) r r rˆ r rˆ( θ, β, µ, α, φ) (A.6) ˆ rˆ ˆ θ θ sinθ i ; cosθ i rˆ θ θ (A.7) rˆ β cos β cosecµ j β ˆ ; ˆ β sin β cos β ˆ β sin β β cos β µ j cos β cos β ( sin β cos ec j ) ec µ ( cos β cos ecµ j ) rˆ µ sin β cos µ cos ec µ j µ (A.8) tan β ˆ β (A.9) ˆ (A.10) ˆ µ sin β sin µ cosec µ j µ sin cos cot cos (A.11) + β µ µ ec µ j ˆ cos µ ( sin β sin µ cosec µ j ) ( ) + cot µ sin β cos µ cos ec µ j µ µ cos µ (A.1) ˆ sin µ ( sin β cos µ cos ec µ j ) ( ) + cot µ sin β cos µ cosec µ j µ µ cos µ ˆ µ tan µ ˆ µ cot µ ˆ µ µ rˆ α cosα cos ecφ j α ˆ ; ˆ α sinα cosecφ j α ˆ α sinα α cosα cosα cosα ( cosα cosecφ j ) rˆ φ sinα cosφ cos ec φ j φ (A.13) ( sinα cosecφ j ) (A.14) (A.15) tanα ˆ α (A.16) ˆ (A.17)
A.E. Eison, E.O. Agbalagba, Johnny A. Francis an Nelson Maxwell 115 ˆ φ sinα sinφ cosec φ j φ ˆ cosφ φ cosφ ( sinα sinφ cosec φ j ) φ ˆ sinφ φ cosφ ( sinα cosφ cosec φ j ) φ + j sinα cosφ cotφ cosec φ (A.18) ( sinα cosφ cosec φ ) (A.19) + cotφ j (A.0) ˆ φ tanφ ˆ φ cotφ ˆ φ (A.1) φ We also now from the rule of ifferentiation that r r r m ( r ) m ( rr. ) m r+ r m r t t t t t r 1 m mr t t (A.) Hence, in orer to remove the factor of which appears in (A.), usually a factor of half is introuce. References [1] E.T. Whittaer, Analytical Dynamics, Cambrige, Englan, Cambrige University Press, pp. 3-76, 1937. [] J.L. Meriam an L.G. Kraige, Engineering mechanics, DYNAMICS, 5 th Eition, pp. 44-89, 009. [3] L. Rosu an C. Haret, Classical Mechanics, Physics Eucation, arxiv.org:physics/9909035. [4] Keith R. Symon, Mechanics, 3r eition, Aison-Wesley Publishing company, pp. 1-76, 1979. [5] Eric W. Weisstein, Central Force, Science Worl, 3, (007), 34-46, http://science worl.wolfram.com/physics/centralforce.html.